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Resolution of Erdős' problems about unimodularity

Stijn Cambie

Abstract

Letting $δ_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$, Erdős wondered if $δ_1(n,m)$ is unimodular for fixed $n$. We prove this is false in general, as the sequence $(δ_1(n,m))$ has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; $n = 1$. We also solve the question on unimodality of the density of integers whose $k^{th}$ prime is $p$.

Resolution of Erdős' problems about unimodularity

Abstract

Letting be the density of the set of integers with exactly one divisor in , Erdős wondered if is unimodular for fixed . We prove this is false in general, as the sequence has superpolynomially many local extrema. However, we confirm unimodality in the single case for which it occurs; . We also solve the question on unimodality of the density of integers whose prime is .
Paper Structure (4 sections, 3 theorems, 8 equations)

This paper contains 4 sections, 3 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Main proofs for problem $692$
  3. Main proofs for problem $690$
  4. First terms of sequences for problem $690$

Key Result

Theorem 1

The sequence $\delta_1(1,m)$ is non-increasing (and thus unimodular) in $m$.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Claim 4
  • proof : Proof
  • Theorem 5
  • proof
  • Claim 6
  • ...and 1 more